Variational and 2D Finite Element Formulations for Size-dependent Elasticity and Piezoelectricity
Darrall, Bradley T.
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Size-dependent elasticity and piezoelectricity variational principles are developed based on recent advances in couple-stress theory and the introduction of an engineering mean curvature vector as energy conjugate to the couple-stresses. It is shown that size-dependent piezoelectricity, sometimes referred to as flexoelectricity, is a straightforward extension of consistent couple-stress elasticity, with electric field and mechanical mean-curvature being thermodynamically coupled. These new variational formulations provide a base for developing couple-stress and size-dependent piezoelectric finite element approaches. By considering the elastic portion of the total potential energy functional to be not only a function of displacement, but of an independent rotation as well, we avoid the necessity to maintain C1 continuity in the finite element methods (FEM) that we develop here. The result is a mixed formulation, which uses Lagrange multipliers to constrain the rotation field to be compatible with the displacement field. Interestingly, this formulation has the noteworthy advantage that the Lagrange multipliers can be shown to be equal to the skew-symmetric part of the force-stress, which otherwise would be cumbersome to calculate. Creating new consistent couple-stress and size-dependent piezoelectric finite element formulations from these variational principles is then a matter of discretizing the variational statement and using appropriate mixed isoparametric elements to represent the domain of interest. The new formulations are then applied to many illustrative examples to bring out important characteristics predicted by consistent couple-stress and size-dependent piezoelectric theories.